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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[ non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b>for <font color="Olive" title="b1">F1</font>, <font color="Olive" title="b2">F2</font> being  <a href="binarith.html#NM1" title="BINARITH:NM.1">Tuple</a> of $1,<a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a>   st  <a href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <font color="Olive" title="b1">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <font color="Olive" title="b2">F2</font> holds <br/><font color="Olive" title="b1">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">F2</font>;<br/>
<a NAME="E1:2"/><i><font color="Green" title="E3">A1</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c1">F1</font>, <font color="Maroon" title="c2">F2</font> be   <a href="binarith.html#NM1" title="BINARITH:NM.1">Tuple</a> of 1,<a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a> ;<br/>

<b>consider </b><font color="Maroon" title="c3">d1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a> <b> such that </b><br/><a NAME="E2:2_1"/><i><font color="Green" title="E4">A2</font></i>: 
<font color="Maroon" title="c1">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c3">d1</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="finseq_2.html#T117" title="FINSEQ_2:th.117">FINSEQ_2:117</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c4">d2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a> <b> such that </b><br/><a NAME="E4:2_1"/><i><font color="Green" title="E5">A3</font></i>: 
<font color="Maroon" title="c2">F2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c4">d2</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="finseq_2.html#T117" title="FINSEQ_2:th.117">FINSEQ_2:117</a></i>;<br/>
<b>assume </b><a NAME="E5:2_1"/><i><font color="Green" title="E6">A4</font></i>: 
 <a href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <font color="Maroon" title="c1">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <font color="Maroon" title="c2">F2</font>
 ;<br/>

<b>assume </b><a NAME="E6:2_1"/><i><font color="Green" title="E7">A5</font></i>: 
<font color="Maroon" title="c1">F1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">F2</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_1_1"/>
( <font color="Maroon" title="c3">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a>  or <font color="Maroon" title="c3">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a>  )
 </span><b>by </b><i><a class="ref" href="xboolean.html#D3" title="XBOOLEAN:def.3">XBOOLEAN:def 3</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_1"><b>suppose </b></a><a NAME="E1:2_1_1_1"/><i><font color="Green" title="E8">A6</font></i>: 
<font color="Maroon" title="c3">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> 
 ;<br/><div class="add"><a NAME="E2:2_1_1_1"/><b>then </b>
<font color="Maroon" title="c4">d2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
 <b>by </b><i><a class="txt" href="binari_3.html#E2:2_1"><i><font color="Green" title="E4">A2</font></i></a>, <a class="txt" href="binari_3.html#E4:2_1"><i><font color="Green" title="E5">A3</font></i></a>, <a class="txt" href="binari_3.html#E6:2_1"><i><font color="Green" title="E7">A5</font></i></a>, <a class="ref" href="xboolean.html#D3" title="XBOOLEAN:def.3">XBOOLEAN:def 3</a></i>;<br/><a NAME="E3:2_1_1_1"/><b>then </b>
(  <a href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <font color="Maroon" title="c1">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <font color="Maroon" title="c2">F2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <b>by </b><i><a class="txt" href="binari_3.html#E2:2_1"><i><font color="Green" title="E4">A2</font></i></a>, <a class="txt" href="binari_3.html#E4:2_1"><i><font color="Green" title="E5">A3</font></i></a>, <a class="txt" href="binari_3.html#E1:2_1_1_1"><i><font color="Green" title="E8">A6</font></i></a>, <a class="ref" href="binarith.html#T41" title="BINARITH:th.41">BINARITH:41</a>, <a class="ref" href="binarith.html#T42" title="BINARITH:th.42">BINARITH:42</a></i>;<br/><b>hence </b><a NAME="E4:2_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="binari_3.html#E5:2_1"><i><font color="Green" title="E6">A4</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2"><b>suppose </b></a><a NAME="E1:2_1_1_2"/><i><font color="Green" title="E8">A7</font></i>: 
<font color="Maroon" title="c3">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
 ;<br/><div class="add"><a NAME="E2:2_1_1_2"/><b>then </b>
<font color="Maroon" title="c4">d2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> 
 <b>by </b><i><a class="txt" href="binari_3.html#E2:2_1"><i><font color="Green" title="E4">A2</font></i></a>, <a class="txt" href="binari_3.html#E4:2_1"><i><font color="Green" title="E5">A3</font></i></a>, <a class="txt" href="binari_3.html#E6:2_1"><i><font color="Green" title="E7">A5</font></i></a>, <a class="ref" href="xboolean.html#D3" title="XBOOLEAN:def.3">XBOOLEAN:def 3</a></i>;<br/><a NAME="E3:2_1_1_2"/><b>then </b>
(  <a href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <font color="Maroon" title="c1">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <a href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <font color="Maroon" title="c2">F2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="txt" href="binari_3.html#E2:2_1"><i><font color="Green" title="E4">A2</font></i></a>, <a class="txt" href="binari_3.html#E4:2_1"><i><font color="Green" title="E5">A3</font></i></a>, <a class="txt" href="binari_3.html#E1:2_1_1_2"><i><font color="Green" title="E8">A7</font></i></a>, <a class="ref" href="binarith.html#T41" title="BINARITH:th.41">BINARITH:41</a>, <a class="ref" href="binarith.html#T42" title="BINARITH:th.42">BINARITH:42</a></i>;<br/><b>hence </b><a NAME="E4:2_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="binari_3.html#E5:2_1"><i><font color="Green" title="E6">A4</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E2:2"/><i><font color="Green" title="E4">A8</font></i>: 
for <font color="Olive" title="b1">k</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c1">k</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:2_2"/><i><font color="Green" title="E5">A9</font></i>: 
for <font color="Olive" title="b1">F1</font>, <font color="Olive" title="b2">F2</font> being  <a href="binarith.html#NM1" title="BINARITH:NM.1">Tuple</a> of <font color="Maroon" title="c1">k</font>,<a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a>   st  <a href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <font color="Olive" title="b1">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <font color="Olive" title="b2">F2</font> holds <br/><font color="Olive" title="b1">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">F2</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c2">F1</font>, <font color="Maroon" title="c3">F2</font> be   <a href="binarith.html#NM1" title="BINARITH:NM.1">Tuple</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a> ;<br/>

<b>consider </b><font color="Maroon" title="c4">T1</font> being   <a href="binarith.html#NM1" title="BINARITH:NM.1">Tuple</a> of <font color="Maroon" title="c1">k</font>,<a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a> , <font color="Maroon" title="c5">d1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a> <b> such that </b><br/><a NAME="E3:2_2"/><i><font color="Green" title="E6">A10</font></i>: 
<font color="Maroon" title="c2">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">T1</font> <a href="binarith.html#K10" title="BINARITH:func.10">^</a> <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c5">d1</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="finseq_2.html#T137" title="FINSEQ_2:th.137">FINSEQ_2:137</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">T2</font> being   <a href="binarith.html#NM1" title="BINARITH:NM.1">Tuple</a> of <font color="Maroon" title="c1">k</font>,<a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a> , <font color="Maroon" title="c7">d2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a> <b> such that </b><br/><a NAME="E5:2_2"/><i><font color="Green" title="E7">A11</font></i>: 
<font color="Maroon" title="c3">F2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">T2</font> <a href="binarith.html#K10" title="BINARITH:func.10">^</a> <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c7">d2</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="finseq_2.html#T137" title="FINSEQ_2:th.137">FINSEQ_2:137</a></i>;<br/>
<b>assume </b><a NAME="E6:2_2"/><i><font color="Green" title="E8">A12</font></i>: 
 <a href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <font color="Maroon" title="c2">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <font color="Maroon" title="c3">F2</font>
 ;<br/>

<i><font color="Green" title="E9">A13</font></i>: <span class="p1">(<span class="default"><a href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <font color="Maroon" title="c4">T1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="funcop_1.html#K15" title="FUNCOP_1:func.15">IFEQ</a> <font color="Maroon" title="c5">d1</font>,<a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c1">k</font></span>)</span></span>)</span> = 
 <a href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <font color="Maroon" title="c2">F1</font>
<b>by </b><i><a class="txt" href="binari_3.html#E3:2_2"><i><font color="Green" title="E6">A10</font></i></a>, <a class="ref" href="binarith.html#T46" title="BINARITH:th.46">BINARITH:46</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <font color="Maroon" title="c6">T2</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="funcop_1.html#K15" title="FUNCOP_1:func.15">IFEQ</a> <font color="Maroon" title="c7">d2</font>,<a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c1">k</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="binari_3.html#E5:2_2"><i><font color="Green" title="E7">A11</font></i></a>, <a class="txt" href="binari_3.html#E6:2_2"><i><font color="Green" title="E8">A12</font></i></a>, <a class="ref" href="binarith.html#T46" title="BINARITH:th.46">BINARITH:46</a></i>
;<br/>
<a NAME="E8:2_2"/>
<font color="Maroon" title="c5">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">d2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:2_2_2"/><i><font color="Green" title="E10">A14</font></i>: 
<font color="Maroon" title="c5">d1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">d2</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_2_2_1"/>
( <font color="Maroon" title="c5">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a>  or <font color="Maroon" title="c5">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a>  )
 </span><b>by </b><i><a class="ref" href="xboolean.html#D3" title="XBOOLEAN:def.3">XBOOLEAN:def 3</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_2_1_1"><b>suppose </b></a><a NAME="E1:2_2_2_1_1"/><i><font color="Green" title="E11">A15</font></i>: 
<font color="Maroon" title="c5">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> 
 ;<br/><div class="add"><a NAME="E2:2_2_2_1_1"/><b>then </b><i><font color="Green" title="E12">A16</font></i>: 
 <a href="funcop_1.html#K15" title="FUNCOP_1:func.15">IFEQ</a> <font color="Maroon" title="c7">d2</font>,<a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c1">k</font>
 <b>by </b><i><a class="txt" href="binari_3.html#E1:2_2_2"><i><font color="Green" title="E10">A14</font></i></a>, <a class="ref" href="funcop_1.html#D8" title="FUNCOP_1:def.8">FUNCOP_1:def 8</a></i>;<br/><a NAME="E3:2_2_2_1_1"/>
 <a href="funcop_1.html#K15" title="FUNCOP_1:func.15">IFEQ</a> <font color="Maroon" title="c5">d1</font>,<a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="binari_3.html#E1:2_2_2_1_1"><i><font color="Green" title="E11">A15</font></i></a>, <a class="ref" href="funcop_1.html#D8" title="FUNCOP_1:def.8">FUNCOP_1:def 8</a></i>;<br/><b>hence </b><a NAME="E4:2_2_2_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="binari_3.html#E7:2_2"><i><font color="Green" title="E9">A13</font></i></a>, <a class="txt" href="binari_3.html#E2:2_2_2_1_1"><i><font color="Green" title="E12">A16</font></i></a>, <a class="ref" href="binari_3.html#T1" target="_self" title="BINARI_3:th.1">Th1</a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_2_1_2"><b>suppose </b></a><a NAME="E1:2_2_2_1_2"/><i><font color="Green" title="E11">A17</font></i>: 
<font color="Maroon" title="c5">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
 ;<br/><div class="add"><a NAME="E2:2_2_2_1_2"/><b>then </b>
<font color="Maroon" title="c7">d2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> 
 <b>by </b><i><a class="txt" href="binari_3.html#E1:2_2_2"><i><font color="Green" title="E10">A14</font></i></a>, <a class="ref" href="xboolean.html#D3" title="XBOOLEAN:def.3">XBOOLEAN:def 3</a></i>;<br/><a NAME="E3:2_2_2_1_2"/><b>then </b><i><font color="Green" title="E12">A18</font></i>: 
 <a href="funcop_1.html#K15" title="FUNCOP_1:func.15">IFEQ</a> <font color="Maroon" title="c7">d2</font>,<a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="funcop_1.html#D8" title="FUNCOP_1:def.8">FUNCOP_1:def 8</a></i>;<br/><a NAME="E4:2_2_2_1_2"/>
 <a href="funcop_1.html#K15" title="FUNCOP_1:func.15">IFEQ</a> <font color="Maroon" title="c5">d1</font>,<a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c1">k</font>
 <b>by </b><i><a class="txt" href="binari_3.html#E1:2_2_2_1_2"><i><font color="Green" title="E11">A17</font></i></a>, <a class="ref" href="funcop_1.html#D8" title="FUNCOP_1:def.8">FUNCOP_1:def 8</a></i>;<br/><b>hence </b><a NAME="E5:2_2_2_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="binari_3.html#E7:2_2"><i><font color="Green" title="E9">A13</font></i></a>, <a class="txt" href="binari_3.html#E3:2_2_2_1_2"><i><font color="Green" title="E12">A18</font></i></a>, <a class="ref" href="binari_3.html#T1" target="_self" title="BINARI_3:th.1">Th1</a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E9:2_2"/>
<font color="Maroon" title="c2">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F2</font>
 <b>by </b><i><a class="txt" href="binari_3.html#E1:2_2"><i><font color="Green" title="E5">A9</font></i></a>, <a class="txt" href="binari_3.html#E3:2_2"><i><font color="Green" title="E6">A10</font></i></a>, <a class="txt" href="binari_3.html#E5:2_2"><i><font color="Green" title="E7">A11</font></i></a>, <a class="txt" href="binari_3.html#E7:2_2"><i><font color="Green" title="E9">A13</font></i></a>, <a class="ref" href="xcmplx_1.html#T2" title="XCMPLX_1:th.2">XCMPLX_1:2</a></i>;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E3:2"/>
for <font color="Olive" title="b1">n</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 <b>from </b><i><a class="ref" href="binarith.html#S1" title="BINARITH:sch.1">BINARITH:sch 1</a>(<a class="txt" href="binari_3.html#E1:2"><i><font color="Green" title="E3">A1</font></i></a>, <a class="txt" href="binari_3.html#E2:2"><i><font color="Green" title="E4">A8</font></i></a>);</i><br/>


</div>
